ON SOME INEQUALITIES AND HANKEL MATRICES INVOLVING PELL, PELL-LUCAS NUMBERS

INVOLVING PELL, PELL-LUCAS NUMBERS

SERPIL HALICI

Communicated by the former editorial board

In [4], the authors defined Toeplitz and Hankel matrices with Pell numbers and gave bounds for the spectral norms of them. In this study, we define Hankel matrices involving the Pell, Pell-Lucas and modified Pell sequences and investigate some properties of them. Moreover, we calculate certain norms of above mentioned matrices. Also, we give upper and lower bounds for spectral norm of Hankel matrix involving Pell, Pell-Lucas, modified Pell numbers.

AMS 2010 Subject Classification: 11B37, 11B39, 15A60.

Key words: recurrence relations, Pell sequences, matrix norms.

1. INTRODUCTION

In the recent years, some considerable works have been done on the norms of some special matrices. In particular, the norms of Toeplitz and Hankel matrices involving Fibonacci and Lucas numbers were investigated in [1] and [2]. In [10], the circulant matrices involving Fibonacci and Lucas numbers were studied and lower and upper bounds for the spectral norms of these matrices were also given. In [4], the authors gave bounds for the spectral norms of Toeplitz and Hankel matrices with Pell numbers. In this study, we define Hankel matrices involving the Pell, Pell-Lucas and modified Pell numbers of the form

A= (aij), aij =Pi+j−1; (1)

B = (bij), bij =Qi+j−1; (2)

C= (c_ij), c_ij =qi+j−1; (3)

respectively, and we calculate the Euclidean, column and row norms involving these numbers. Also, we give bounds for the spectral norms of the matrices involving Pell, Pell-Lucas and modified Pell numbers, that is, the bounds of spectral norm related to these numbers. We find a new formula which is different from the formula given in the reference [4], which refers to matrices

MATH. REPORTS15(65),1 (2013), 1–10

with different indexing. And this formula involves not only Pell numbers but also Pell-Lucas, modified Pell numbers.

Now, we give some fundamental knowledge related to our study. The Pell sequence is defined by the following recursive relation, for n≥2

(4) P_n= 2Pn−1+Pn−2,

where P0 = 0 and P1 = 1 [3]. And the Pell-Lucas sequence is

(5) Q_n= 2Qn−1+Qn−2,

where Q0 = 2 and Q1 = 2. Similarly, the modified Pell sequence is defined as follows.

(6) q_n= 2qn−1+qn−2, q₀ = 1 =q₁.

Extension to negative values ofnmay be made, but here we do not care about it. If we start from n= 0, then the first elements of the Pell, Pell-Lucas and modified Pell sequences are given by

n 0 1 2 3 4 5 6 7 8 9 · · ·

P_n 0 1 2 5 12 29 70 169 408 985 · · · Qn 2 2 6 14 34 82 198 478 1154 2786 · · · q_n 1 1 3 7 17 41 99 239 577 1393 · · ·

Some important elementary relationships involvingPn,Qnandqnfollow without difficulty with the aid of the Binet formulas.

(7)

n

X

k=1

P_k² = P_nP_n+1

2 ,

(8)

n−1

X

k=1

P_kP_k+1 = P2n+1−2Pn+1Pn−1

4 ,

(9) 2Pn−1Pn+P_n−1² −P_n²= (−1)ⁿ, (10) P2n+1=P_n²+P_n+1² ,

(11) P_n²+P_n+1Pn−1 = 1

4Q²_n,

(12) P_n² = 1

8(Q_2n−2 (−1)ⁿ), (13)

n

X

k=1

Q²_k= Q_2n+1+ 2 (−1)ⁿ−4

2 ,

n

X

k=1

Q_2k+1 = Q_2n+2−6

2 ,

(14) (Pn+1−Pn)² = 2P_n²+ (−1)ⁿ,

(15) 4Pn=Qn+Qn−1, Pn+1−Pn=qn, Pn−1+Pn=qn. The Hankel matrix is an n×n matrix

Hn= (hij)ⁿ_i,j=1, where h_ij =hi+j−1, i.e., a matrix of the form

H_n=



















h1 h2 h3 · · · hn−1 hn

h₂ h₃ h₄ · · · h_n h_n+1 h3 h4 h5 · · · hn+1 hn+2

... ... ... . .. ... ... hn−1 hn hn+1 · · · h2n−3 h2n−2

h_n h_n+1 h_n+2 · · · h2n−2 h2n−1

















 .

LetA= (a_ij) be an m×n matrix. The Frobenius or Euclidean norm of A is defined as

kAk_F =





m

X

i=1 n

X

j=1

|a_ij|²





1 2

and also the spectral norm of A is kAk₂ =

r

1≤i≤nmax |λ_i|,

respectively, where the numbers λ_i are the eigenvalues of matrix A^HA. The matrix A^H is the conjugate transpose of the matrixA. The column and row norm of matrix A are defined by

kAk₁= max

1≤j≤n m

X

i=1

|a_ij|, kAk_∞= max

1≤i≤m n

X

j=1

|a_ij|,

respectively [8]. For the matrices A= (aij)_m×n and B = (bij)_m×n, the Hada- mard product of these matrices is defined as

A◦B = (a_ijb_ij).

For the m×n matrixA, the following inequality is satisfied (see [8]).

(16) 1

pmin(m, n)kAk_F ≤ kAk₂ ≤ kAk_F.

The maximum column length normc₁(·) and maximum row length normr₁(·) for the matrix A= (aij)_m×n are defined by

(17) c1(A)≡max

j

s X

i

|a_ij|²= max

j

[aij]^m_i=1 F,

and

(18) r1(A)≡max

i

s X

j

|a_ij|² = max

i

[aij]ⁿ_j=1 F,

respectively.

In [7], the author showed that ifA◦B =C, then

(19) kCk₂≤r₁(A)c₁(B).

2. MAIN THEOREMS

In this section, we give the Euclidean, spectral, column and row norms of the Hankel matrices with Pell, Pell-Lucas and modified Pell numbers.

Theorem 1. If A is an n×n matrix A= (a_ij) with a_ij =Pi+j−1, then we have

kAk_F = 1 2

q

P_2n² −2P_n²+ξ, or equivalently

kAk_F = 1 2√

2 q

q_2n² −2q²_n+ 1,

where k·k_F is the Frobenius norm, Pn denotes the nth Pell number and ξ =

2 if n odd, 0 if n even.

Proof. Since

A=



















P₁ P₂ P₃ · · · Pn−1 P_n P₂ P₃ P₄ · · · P_n P_n+1 P3 P4 P5 · · · Pn+1 Pn+2

... ... ... . .. ... ... Pn−1 Pn Pn+1 · · · P2n−3 P2n−2

P_n P_n+1 P_n+2 · · · P2n−2 P2n−1



















and

n

X

k=1

P_k²= P_nP_n+1

2 ,

n−1

X

k=1

P_kP_k+1 = P_2n+1−2P_nP_n+1−1

4 ,

we have kAk_F =





n

X

i=1 n

X

j=1

|a_ij|²





1 2

=

n

X

k=1

P_k²+

n+1

X

k=2

P_k²+· · ·+

2n−1

X

k=n

P_k²

!¹₂ ,

kAk_F = _n

X

k=1

P_k²+

n+1

X

k=1

P_k²+· · ·+

2n−1

X

k=1

P_k²

!

−

n−1

X

k=1 k

X

i=1

P_i²

!!¹₂ ,

kAk_F = 1

2(PnPn+1+Pn+1Pn+2+· · ·+P2n−1P2n)−1 2

n−1

X

k=1

P_kP_k+1

!¹₂ ,

kAk_F = 1 2

2n−1

X

k=1

PkPk+1−2

n−1

X

k=1

PkPk+1

!!¹₂ ,

kAk_F = P_2n² +P_2n+1² −2P_2nP_2n+1−2 P_n²+P_n+1² −2P_nP_n+1 + 1 8

!¹₂ ,

kAk_F = 1

2 P_2n² −2P_n²+ 1−(−1)ⁿ¹₂ .

Thus, we obtain the desired equality. Likewise, from the equationPn+1−P_n= qn, it can be seen that

kAk_F = 1 2√

2 q

q_2n² −2q²_n+ 1.

Thus, the proof is completed.

In the following theorem, the column and row norms of the matrixAare given in terms of the modified Pell numbers.

Theorem 2. Let A be ann×nmatrix witha_ij =Pi+j−1. Then we have kAk₁=kAk_∞= 1

2(q2n−qn),

where kAk₁, kAk_∞ are the column and row norms, respectively.

Proof. From the definition of the matrix A, we can write kAk₁ = max

1≤j≤n n

X

i=1

|a_ij|= max

1≤j≤n{|a_1j|+|a_2j|+· · ·+|a_nj|}, kAk₁ =P_n+P_n+1+· · ·+P2n−1,

kAk₁ =

2n−1

X

i=1

P_i−

n−1

X

i=1

P_i.

Thus, we get

kAk₁ = (P_2n+P2n−1−1)−(P_n+Pn−1−1)

2 = 1

2(q_2n−q_n).

Similarly, the row norm of matrixAcan be computed as kAk_∞= max

1≤i≤n n

X

j=1

|a_ij|=

2n−1

X

k=n

Pk = 1

2(q2n−qn), as desired.

Now, we give bounds for spectral norm of the matrix A involving Pell and modified Pell numbers. Note that this bound is different from the one in [4] because there the indexing starts from 0, while here it starts from 1.

Theorem 3. If A is an n×n matrix A= (aij) with aij =Pi+j−1, then we have

1 2√

n q

P_2n² −2P_n²+ξ≤

≤ kAk₂ ≤ 1 2

p(P2nP2n−1−PnPn−1) (P2n−2P2n−1−PnPn−1+ 2) and

1 2√

2n q

q_2n² −2q²_n+ 1≤

≤ kAk₂ ≤ 1 8

p(q4n−1−q2n−1+ξ) (q4n−3−q2n−1+ξ+ 8).

Proof. From Theorem 1 and the inequality (16), we can write 1

2√ n

q

P_2n² −2P_n²+ξ ≤ kAk₂ and 1 2√

2n q

q²_2n−2q_n²+ 1≤ kAk₂. On the other hand, let us define two new matrices Un= (uij)ⁿ_i,j=1, where

u_ij =

Pi+j−1 i≤j, 1 i > j, and V_n= (v_ij)ⁿ_i,j=1, where

vij =

Pi+j−1 i > j,

1 i≤j.

That is, we write

Un= (uij) =



















P1 P2 P3 · · · Pn−1 Pn

1 P3 P4 · · · Pn Pn+1

1 1 P₅ · · · P_n+1 P_n+2 ... ... ... . .. ... ... 1 1 1 · · · P2n−3 P2n−2

1 1 1 · · · 1 P2n−1



















and

V_n= (v_ij) =















1 1 1 · · · 1

P₂ 1 1 · · · 1 P3 P4 1 · · · 1 ... ... ... . .. ...

P_n P_n+1 P_n+2 · · · 1













 ,

respectively. It can be easily seen that A=Un◦V_n . Thus, we obtain that r₁(U_n) = max

i

s X

j

|u_ij|² = v u u t

2n−1

X

i=n

P_i² =

= v u u t

2n−1

X

i=1

P_i²−

n−1

X

i=1

P_i² =

rP2n−1P_2n−Pn−1P_n 2

and

c1(Vn) = max

j

s X

i

|v_ij|² = r

1 +P2n−2P2n−1−Pn−1Pn

2 .

Then, using the inequality kCk₂ ≤r₁(A)c₁(B) we get (20) kAk₂≤ 1

2

p(P2nP2n−1−PnPn−1) (P2n−2P2n−1−PnPn−1+ 2).

If we use the inequality (20) and the inequality ₂^√1_np

P_2n² −2P_n²+ξ≤ kAk₂, then we obtain that

1 2√ n

q

P_2n² −2P_n²+ξ≤ kAk₂ ≤

≤ 1 2

p(P_2nP2n−1−P_nPn−1) (P2n−2P2n−1−P_nPn−1+ 2).

In a similar way the other inequality can be easily seen. Thus, the proof is completed.

In the following theorem, we give the Euclidean (Frobenius) norm of the matrix involving Pell-Lucas numbers.

Theorem 4. If B is an n×n matrix with b_ij =Qi+j−1, then we have kBk_F = 1

2 q

Q_4n−2Q_2n−2 + 4 (−1)ⁿ, or

kBk_F = q

2 P_2n² −2P_n² ,

where k·k_F is the Frobenius norm and Qn denotes thenth Pell-Lucas number.

Proof. From the definition of Frobenius norm we can write the following equations for the matrix B

kBk_F =





n

X

i=1 n

X

j=1

|b_ij|²





1 2

=

n

X

k=1

Q²_k+

n+1

X

k=2

Q²_k+· · ·+

2n−1

X

k=n

Q²_k

!¹₂ ,

kBk_F = _n

X

k=1

Q²_k+

n+1

X

k=1

Q²_k+· · ·+

2n−1

X

k=1

Q²_k

!

−

n−1

X

k=1 k

X

i=1

Q²_i

!!¹₂ .

If we use the equations given in (13), then we obtain kBk_F = 1

2

"

Q_4n−6

2 + 2−1−(−1)²ⁿ

2 + 41 + (−1)ⁿ

2 −Q_2n+ 6−4

#!¹₂ . That is,

kBk_F = 1 2

q

Q4n−2Q2n−2 + 4 (−1)ⁿ.

Similarly, if we use the equations Pn+Pn−1 =qn= ^Q₂ⁿ then we have kBk_F =

q

2 P_2n² −2P_n² . So, the proof is completed.

The following corollary determines the column and row norms of matrix B and its proof can be easily seen.

Corollary 1. IfB is ann×nmatrix withbij =Qi+j−1, then we have kBk₁ =kBk_∞= 1

2(Q2n+Q2n−1−Qn−Qn−1) = 2 (P2n−Pn). Theorem 5. If B is an n×n matrix with b_ij =Qi+j−1, then we have

1 2√

n q

Q4n−2Q2n−2 + 4 (−1)ⁿ≤ kBk₂≤

≤ 1 2

p(Q4n−Q2n) (Q4n−2−Q2n+ 2), and

r2

n(P_2n² −2P_n²)≤ kBk₂ ≤ q

4(P_2n² −P_n²) +ξ

4(P_2n−1² −P_n²) + (−1)ⁿ⁺¹ . Proof. Let us define two matrices R_n = (r_ij)ⁿ_i,j=1 and S_n = (s_ij)ⁿ_i,j=1 such that

r_ij =

Qi+j−1 i≤j, 1 i > j

and

sij =

Qi+j−1 i > j,

1 i≤j,

respectively. Using the inequality kCk₂ ≤r1(A)c1(B),we can write kBk₂≤r1(Rn)c1(Sn).

From the definitions ofr1(Rn) and c1(Sn) we have that

r1(Rn) = max

i

s X

j

|r_ij|² = v u u t

n

X

j=1

|r_nj|²,

r₁(R_n) = v u u t

2n−1

X

i=n

Q²_i = v u u t

2n−1

X

i=1

Q²_i −

n−1

X

i=1

Q²_i =

rQ_4n−Q_2n

2 ,

and

c₁(S_n) = max

j

s X

i

|s_ij|²= v u u t1 +

2n−2

X

i=n

Q²_i =

rQ4n−2−Q_2n+ 2

2 .

Thus, the upper bound of B can be found as kBk₂ ≤ 1

2

p(Q_4n−Q_2n) (Q4n−2−Q_2n+ 2).

On the other hand, using the inequality ^√1_nkAk_F ≤ kAk₂ ≤ kAk_F,we get r2

n P_2n² −2P_n²

≤ kBk₂≤q

4 P_2n² −P_n² +ξ

4 P_2n−1² −P_n²

+ (−1)ⁿ⁺¹ . Thus, the proof is completed.

The following corollary is obvious if the norm axiomkαAk=|α| kAkand the identity Q_n= 2q_n are used.

Corollary 2. For the matrices C = (cij)n×n, cij = qi+j−1, and B = (b_ij)n×n, b_ij =Qi+j−1, we have the following results:

kCk_F = 1 2√

2 q

q_4n−q_2n−1 + 2 ( −1)ⁿ or kCk_F = r1

2 P_2n² −P_n² . (i)

kCk₁=kCk_∞= 1

2(q_2n+q2n−1−q_n−qn−1) =P_2n−P_n. (ii)

1 2√

2n

pq_4n−2q_2n−1+2(−1)ⁿ≤ kCk₂≤p

(q_4n−q_2n)(q_4n−2−q_2n+1).

(iii)

√1 2n

q

P_2n² −2P_n²

≤ kBk₂ ≤ (iv)

≤ 1 2

r

4 P_2n² −P_n² +ξ

4 P_2n−1² −P_n²

+ (−1)ⁿ⁺¹ .

Acknowledgements.The author is very grateful to an anonymous referee for helpful comments and suggestions to improve the presentation of this paper.

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Received 13 July 2011 Sakarya University

Faculty of Sciences Department of Mathematics

54187 Sakarya, Turkey [email protected]